Question

More Confidence Intervals For the confidence intervals given in Exercises \(7-10,\) can you conclude that there is a difference between \(\mu_{1}\) and \(\mu_{2}\) ? Explain. $$136.2<\mu_{1}-\mu_{2}<137.3$$

Short answer

Answer: Yes, there is a significant difference between the two population means, as the confidence interval does not include zero.

Step-by-step solution

Examine the given confidence interval

We have the following confidence interval for the difference of population means: $$136.2<\mu_{1}-\mu_{2}<137.3$$

Check if the interval includes zero

Observe the given confidence interval: $$136.2<\mu_{1}-\mu_{2}<137.3$$ This interval does not include zero as all values are greater than zero. So, no value within the range suggests that there would be no difference between \(\mu_{1}\) and \(\mu_{2}\).

Conclusion

Since the confidence interval does not include zero, it concludes that there is a significant difference between the two population means \(\mu_{1}\) and \(\mu_{2}\).

Key concepts

Difference of Population Means

Understanding the difference of population means is crucial in statistics, especially when comparing distinct groups. In the given exercise, the difference between two population means, \(\mu_1\) and \(\mu_2\), is considered. The concept can be visualized as the distance or gap between two averages from different groups. When a confidence interval is used, it provides us with a range in which we believe the true difference in population means lies based on sample data.

For example, if the average height of adult men in two different countries has a certain gap, with a confidence interval between 136.2 and 137.3 cm, we are fairly certain that the true difference in average heights is within this range. The implication here is that one population is consistently taller than the other. It is essential that the chosen confidence level (commonly 95% or 99%) affects both the width of this interval and our certainty in this estimation.

Statistical Significance

The term 'statistical significance' represents the likelihood that the difference observed between two groups is not due to sampling error. In simpler terms, it means that the findings are meaningful and not just a random occurrence. In the exercise, since zero is not in the confidence interval (\(136.2<\mu_{1}-\mu_{2}<137.3\)), we can say that the difference in means is statistically significant.

• If the interval had included zero, it would suggest that the difference could be zero, implying no statistical significance.
• Excluding zero implies that the likelihood of there being no difference is very low – hence, the difference observed is statistically significant.

This is critical in research for validating hypotheses about population parameters. When a difference is statistically significant, researchers and analysts can proceed with a higher degree of confidence in their findings.

Hypothesis Testing

Hypothesis testing is a method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It begins with an initial hypothesis, known as the null hypothesis (\(H_0\)), which states there is no effect or no difference, and the alternative hypothesis (\(H_A\)), which states there is an effect or a difference.

To test these, statisticians consider the p-value, which measures the probability of observing the data or something more extreme, given that the null hypothesis is true. If the p-value is less than the chosen threshold (commonly \(\alpha = 0.05\)), the null hypothesis is rejected.
\[\begin{equation} p\text{-value} < \alpha \Rightarrow \text{Reject } H_0 \end{equation}\]
This conclusion is supported by our confidence interval since it does not include zero, leading us to reject the null hypothesis in favor of the alternative, which suggests that there is a true difference between the population means.